Can someone help me understand how this was done with the armature?
I try to understand your problem. If you're talking about, let say, the left index finger controller, you can see on the N panel that its location and the X and Z scales are locked: Actually he uses both Y SCALE and ROTATION to make the finger bend, so its movement completely depend on these constraints. If you want this finger to do something else you need to free it from these constraints on the particular frame, or all along the timeline. I personally find it a bit complicated but maybe he has good reasons to rig this way.